Multi-scale finite-volume method for use in subsurface flow simulation

ABSTRACT

A multi-scale finite-volume (MSFV) method to solve elliptic problems with a plurality of spatial scales arising from single or multi-phase flows in porous media is provided. Two sets of locally computed basis functions are employed. A first set of basis functions captures the small-scale heterogeneity of the underlying permeability field, and it is computed to construct the effective coarse-scale transmissibilities. A second set of basis functions is required to construct a conservative fine-scale velocity field. The method efficiently captures the effects of small scales on a coarse grid, is conservative, and treats tensor permeabilities correctly. The underlying idea is to construct transmissibilities that capture the local properties of a differential operator. This leads to a multi-point discretization scheme for a finite-volume solution algorithm. Transmissibilities for the MSFV method are preferably constructed only once as a preprocessing step and can be computed locally. Therefore, this step is well suited for massively parallel computers. Furthermore, a conservative fine-scale velocity field can be constructed from a coarse-scale pressure solution which also satisfies the proper mass balance on the fine scale. A transport problem is ideally solved iteratively in two stages. In the first stage, a fine scale velocity field is obtained from solving a pressure equation. In the second stage, the transport problem is solved on the fine cells using the fine-scale velocity field. A solution may be computed on the coarse cells at an incremental time and properties, such as a mobility coefficient, may be generated for the fine cells at the incremental time. If a predetermined condition is not met for all fine cells inside a dual coarse control volume, then the dual and fine scale basis functions in that dual coarse control volume are reconstructed.

TECHNICAL FIELD

The present invention relates generally to subsurface reservoirsimulators, and more particularly, to those simulators which usemulti-scale physics to simulate flow in an underground reservoir.

BACKGROUND OF THE INVENTION

The level of detail available in reservoir description often exceeds thecomputational capability of existing reservoir simulators. Thisresolution gap is usually tackled by upscaling the fine-scaledescription to sizes that can be treated by a full-featured simulator.In upscaling, the original model is coarsened using a computationallyinexpensive process. In flow-based methods, the process is based onsingle-phase flow. A simulation study is then performed using thecoarsened model. Upscaling methods such as these have proven to be quitesuccessful. However, it is not possible to have a priori estimates ofthe errors that are present when complex flow processes are investigatedusing coarse models constructed via these simplified settings.

Various fundamentally different multi-scale approaches for flow inporous media have been proposed to accommodate the fine-scaledescription directly. As opposed to upscaling, the multi-scale approachtargets the full problem with the original resolution. The upscalingmethodology is typically based on resolving the length and time-scalesof interest by maximizing local operations. Arbogast et al. (T.Arbogast, Numerical subgrid upscaling of two phase flow in porous media,Technical report, Texas Institute for Computational and AppliedMathematics, The University of Texas at Austin, 1999, and T. Arbogastand S. L. Bryant, Numerical subgrid upscaling for waterfloodsimulations, SPE 66375, 2001) presented a mixed finite-element methodwhere fine-scale effects are localized by a boundary conditionassumption at the coarse element boundaries. Then the small-scaleinfluence is coupled with the coarse-scale effects by numerical Greensfunctions. Hou and Wu (T. Hou and X. H. Wu, A multiscale finite elementmethod for elliptic problems in composite materials and porous media, J.Comp. Phys., 134:169-189, 1997) employed a finite-element approach andconstructed specific basis functions which capture the small scales.Again, localization is achieved by boundary condition assumptions forthe coarse elements. To reduce the effects of these boundary conditions,an oversampling technique can be applied. Chen and Hou (Z. Chen and T.Y. Hou, A mixed finite element method for elliptic problems with rapidlyoscillating coefficients, Math. Comput., June 2002) utilized these ideasin combination with a mixed finite-element approach. Another approach byBeckie et al. (R. Beckie, A. A. Aldama, and E. F. Wood, Modeling thelarge-scale dynamics of saturated groundwater flow using spatialfiltering, Water Resources Research, 32:1269-1280, 1996) is based onlarge eddy simulation (LES) techniques which are commonly used forturbulence modeling.

Lee et al. (S. H. Lee, L. J. Durlofsky, M. F. Lough, and W. H. Chen,Finite difference simulation of geologically complex reservoirs withtensor permeabilities, SPERE&E, pages 567-574, 1998) developed aflux-continuous finite-difference (FCFD) scheme for 2D models. Lee etal. further developed a method to address 3D models (S. H. Lee, H.Tchelepi, P. Jenny and L. Dechant, Implementation of a flux continuousfinite-difference method for stratigraphic, hexahedron grids, SPEJournal, September, pages 269-277, 2002). Jenny et al. (P. Jenny, C.Wolfsteiner, S. H. Lee and L. J. Durlofsky, Modeling flow ingeometrically complex reservoirs using hexahedral multi-block grids, SPEJournal, June, pages 149-157, 2002) later implemented this scheme in amulti-block simulator.

In light of the above modeling efforts, there is a need for a simulationmethod which more efficiently captures the effects of small scales on acoarse grid. Ideally, the method would be conservative and also treattensor permeabilities correctly. Further, preferably the reconstructedfine-scale solution would satisfy the proper mass balance on thefine-scale. The present invention provides such a simulation method.

SUMMARY OF THE INVENTION

A multi-scale finite-volume (MSFV) approach is taught for solvingelliptic or parabolic problems such as those found in subsurface flowsimulators. Advantages of the present MSFV method are that it fitsnicely into a finite-volume framework, it allows for computing effectivecoarse-scale transmissibilities, treats tensor permeabilities properly,and is conservative at both the coarse and fine scales. The presentmethod is computationally efficient relative to reservoir simulation nowin use and is well suited for massive parallel computation. The presentinvention can be applied to 3D unstructured grids and also tomulti-phase flow. Further, the reconstructed fine-scale solutionsatisfies the proper mass balance on the fine-scale.

A multi-scale approach is described which results in effectivetransmissibilities for the coarse-scale problem. Once thetransmissibilities are constructed, the MSFV method uses a finite-volumescheme employing multi-point stencils for flux discretization. Theapproach is conservative and treats tensor permeabilities correctly.This method is easily applied using existing finite-volume codes, andonce the transmissibilities are computed, the method is computationallyvery efficient. In computing the effective transmissibilities, closureassumptions are employed.

A significant characteristic of the present multi-scale method is thattwo sets of basis functions are employed. A first set of dual basisfunctions is computed to construct transmissibilities between-coarsecells. A second set of locally computed fine scale basis functions isutilized to reconstruct a fine-scale velocity field from a coarse scalesolution. This second set of fine-scale basis functions is designed suchthat the reconstructed fine-scale velocity solution is fully consistentwith the transmissibilities. Further, the solution satisfies the propermass balance on the small scale.

The MSFV method may be used in modeling a subsurface reservoir. A finegrid is first created defining a plurality of fine cells. A permeabilityfield and other fine scale properties are associated with the finecells. Next, a coarse grid is created which defines a plurality ofcoarse cells having interfaces between the coarse cells. The coarsecells are ideally aggregates of the fine cells. A dual coarse grid isconstructed defining a plurality of dual coarse control volumes. Thedual coarse control volumes are ideally also aggregates of the finecells. Boundaries surround the dual coarse control volumes.

Dual basis functions are then calculated on the dual coarse controlvolumes by solving local elliptic or parabolic problems, preferablyusing boundary conditions obtained from solving reduced problems alongthe interfaces of the course cells. Fluxes, preferably integral fluxes,are then extracted across the interfaces of the coarse cells from thedual basis functions. These fluxes are assembled to obtain effectivetransmissibilities between coarse cells of the coarse cell grid. Thetransmissibilities can be used for coarse scale finite volumecalculations.

A fine scale velocity field may be established. A finite volume methodis used to calculate pressures in the coarse cells utilizing thetransmissibilities between cells. Fine scale basis functions arecomputed by solving local elliptic or parabolic flow problems on thecoarse cells and by utilizing fine scale fluxes across the interfaces ofthe coarse cells which are extracted from the dual basis functions.Finally, the fine-scale basis functions and the corresponding coarsecell pressures are combined to extract the small scale velocity field.

A transport problem may be solved on the fine grid by using the smallscale velocity field. Ideally, the transport problem is solvediteratively in two stages. In the first stage, a fine scale velocityfield is obtained from solving a pressure equation. In the second stage,the transport problem is solved on the fine cells using the fine-scalevelocity field. A Schwartz overlap technique can be applied to solve thetransport problem locally on each coarse cell with an implicit upwindscheme.

A solution may be computed on the coarse cells at an incremental timeand properties, such as a mobility coefficient, may be generated for thefine cells at the incremental time. If a predetermined condition is notmet for all fine cells inside a dual coarse control volume, then thedual and fine scale basis functions in that dual coarse control volumeare reconstructed.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other objects, features and advantages of the presentinvention will become better understood with regard to the followingdescription, pending claims and accompanying drawings where:

FIG. 1 illustrates a coarse 2D grid of coarse cells with an overlyingdual coarse grid including a dual coarse control volume and anunderlying fine grid of fine cells;

FIG. 2 illustrates a coarse grid including nine adjacent coarse cells(bold solid lines) with a corresponding overlying dual coarse grid (bolddashed lines) including dual coarse control volumes and an underlyingfine grid (thin dotted lines) of fine cells;

FIG. 3 shows flux contribution q_(A) ⁽²⁾ and q_(B) ⁽²⁾ due to thepressure in a particular coarse cell 2;

FIG. 4 is a flowchart describing the overall steps used in a preferredembodiment of a reservoir simulation which employs a multi-scalefinite-volume (MSFV) method made in accordance with this invention;

FIG. 5 is a flowchart further detailing steps used to determinetransmissibilities T between coarse cells;

FIG. 6 is a flow chart further describing steps used to construct a setof fine-scale basis functions and to extract a small scale velocityfield;

FIG. 7 is a flowchart depicting coupling between pressure and thesaturation equations which utilize an implicit solution scheme andwherein Π and Σ are operators used to update total velocity andsaturation, respectively, during a single time step;

FIG. 8 is an illustration of the use of an adaptive scheme toselectively update basis functions;

FIG. 9 is an illustration of a permeability field associated with a SPE10 problem;

FIGS. 10A-B are illustrations of permeability fields of a top layer anda bottom layer of cells from the SPE 10 problem;

FIGS. 11A-B are illustrations of saturation fields of top layers ofcells created using the MSFV method and FIG. 11C is an illustration of asaturation field computed by a conventional fine-scale reservoirsimulator;

FIGS. 12A-B are illustrations of saturation fields of bottom layers ofcells created using the MSFV method and FIG. 12C is an illustration of asaturation field computed by a conventional fine-scale reservoircomputer;

FIGS. 13A-B are graphs of oil cut and oil recovery;

FIG. 14 is an illustration of a 3D test case having a grid of 10×22×17grid cells and including injector and producer wells; and

FIG. 15 is a graph of oil cut and oil recovery.

BEST MODES FOR CARRYING OUT THE INVENTION I. Flow Problem

A. One Phase Flow

Fluid flow in a porous media can be described by the elliptic problem:∇·(λ·∇p)=ƒon Ω  (1)where p is the pressure, λ is the mobility coefficient (permeability, K,divided by fluid viscosity, u) and Ω is a volume or region of asubsurface which is to be simulated. A source term ƒ represents wells,and in the compressible case, time derivatives. Permeabilityheterogeneity is a dominant factor in dictating the flow behavior innatural porous formations. The heterogeneity of permeability K isusually represented as a complex multi-scale function of space.Moreover, permeability K tends to be a highly discontinuous full tensor.Resolving the spatial correlation structures and capturing thevariability of permeability requires a highly detailed reservoirdescription.

The velocity u of fluid flow is related to the pressure field throughDarcy's law:u=−λ·∇p  (2)

On the boundary of a volume, ∂Ω, the flux q=u·v is specified, where v isthe boundary unit normal vector pointing outward. Equations (1) and (2)describe incompressible flow in a porous media. These equations applyfor both single and multi-phase flows when appropriate interpretationsof the mobility coefficient λ and velocity u are made. This ellipticproblem is a simple, yet representative, description of the type ofsystems that should be handled efficiently by a subsurface flowsimulator. Moreover, the ability to handle this limiting case ofincompressible flow ensures that compressible systems can be treated asa subset.

B. Two Phase Flow

The flow of two incompressible phases in a heterogeneous domain may bemathematically described by the following: $\begin{matrix}\begin{matrix}{{{\Phi\quad\frac{\partial S_{o}}{\partial t}} - {\frac{\partial}{\partial x_{i}}\left( {k\quad\frac{k_{r_{o}}}{\mu_{o}}\frac{\partial p}{\partial x_{i}}} \right)}} = {- q_{o}}} \\{{{\Phi\quad\frac{\partial S_{w}}{\partial t}} - {\frac{\partial}{\partial x_{i}}\left( {k\quad\frac{k_{r_{w}}}{\mu_{w}}\frac{\partial p}{\partial x_{i}}} \right)}} = {- q_{w}}}\end{matrix} & (3)\end{matrix}$on a volume Ω, where p is the pressure, S_(o,w) are the saturations (thesubscripts o and w stand for oil and water, respectively) with0≦S_(o,w)≦1 and S_(o)+S_(w)≡1, k is the heterogeneous permeability,k_(r) _(o,w) are the relative permeabilities (which are functions ofS_(o,w)), u_(o,w) the viscosities and q_(o,w) are source terms whichrepresent the wells. The system assumes that capillary pressure andgravity are negligible. Equivalently, system (3) can be written as:−∇·u=q _(o) +q _(w)  (4)$\begin{matrix}{{{\Phi\quad\frac{\partial S_{o}}{\partial t}} + {\nabla{\cdot \left( {\frac{k_{o}}{k_{o} + k_{w}}u} \right)}}} = {- q_{o}}} & (5)\end{matrix}$on Ω withu=−λ∇p.  (6)and the total mobilityλ=k(k _(o) +K _(w)),  (7)where k_(j)≡k_(rj)/u_(j) for jε{o,w}.

Equation (4) is known as the “pressure equation” and equation (5) as the“hyperbolic transport equation.” Again, equations (4) and (5) are arepresentative description of the type of systems that should be handledefficiently by a subsurface flow simulator. Such flow simulators, andtechniques employed to simulate flow, are well known to those skilled inthe art and are described in publications such as Petroleum ReservoirSimulation, K. Aziz and A. Settari, Stanford Bookstore CustomPublishing, 1999.

II. Multi-Scale Finite-Volume (MSFV) Method

A. MSFV Method for One Phase Flow

1. Finite-Volume Method

A cell centered finite-volume method will now be briefly described. Tosolve the problem of equation (1), the overall domain or volume Ω ispartitioned into smaller volumes {{overscore (Ω)}_(i)}. A finite-volumesolution then satisfies $\begin{matrix}{{\int_{{\overset{\_}{\Omega}}_{i}}^{\quad}{{\nabla{\cdot u}}\quad{\mathbb{d}\Omega}}} = {{\int_{\partial{\overset{\_}{\Omega}}_{i}}^{\quad}{{u \cdot \overset{\_}{v}}\quad{\mathbb{d}\Gamma}}} = {- {\int_{{\overset{\_}{\Omega}}_{i}}^{\quad}{f\quad{\mathbb{d}\Omega}}}}}} & (8)\end{matrix}$for each control volume {overscore (Ω)}_(i), where {overscore (v)} isthe unit normal vector of the volume boundary ∂{overscore (Ω)}_(i)pointing outward. The challenge is to find a good approximation foru·{overscore (v)} at ∂{overscore (Ω)}_(i). In general, the flux isexpressed as: $\begin{matrix}{{u \cdot \overset{\_}{v}} = {\sum\limits_{k = 1}^{n}{T^{k}{{\overset{\_}{p}}^{k}.}}}} & (9)\end{matrix}$

Equation (9) is a linear combination of the pressure values, {overscore(p)}, in the volumes {{overscore (Ω)}_(i)} of the domain Ω. The totalnumber of volumes is n and T^(k) denotes transmissibility betweenvolumes {{overscore (Ω)}_(i)}. By definition, the fluxes of equation (9)are continuous across the interfaces of the volumes {{overscore(Ω)}_(i)} and, as a result, the finite-volume method is conservative.

2. Construction of the Effective Transmissibilities

The MSFV method results in multi-point stencils for coarse-scale fluxes.For the following description, an orthogonal 2D grid 20 of grid cells 22is used, as shown in FIG. 1. An underlying fine grid 24 of fine gridcells 26 contains the fine-scale permeability K information. To computethe transmissibilities T between coarse grid cells 22, a dual coarsegrid 30 of dual coarse control volumes 32 is used. A control volume 32of the dual grid 30, {tilde over (Ω)}, is constructed by connecting themid-points of four adjacent coarse grid cells 22. To relate the fluxesacross the coarse grid cell interfaces 34 which lie inside a particularcontrol volume 32, or {tilde over (Ω)}, to the finite-volume pressures{overscore (p)}^(k)(k=1,4) in the four adjacent coarse grid cells 22, alocal elliptical problem in the preferred embodiment is defined as∇·(λ·∇p)=0on {tilde over (Ω)}.  (10)

For one skilled in the art, the method can easily be adapted to use alocal parabolic problem.

For an elliptic problem, Dirichlet or Neumann boundary conditions are tobe specified on boundary ∂{tilde over (Ω)}. Ideally, the imposedboundary conditions should approximate the true flow conditionsexperienced by the sub-domain in the full system. These boundaryconditions can be time and flow dependent. Since the sub-domain isembedded in the whole system, Wallstrom et al. (T. C. Wallstrom, T. Y.Hou, M. A Christie, L. J. Durlofsky, and D. H. Sharp, Application of anew two-phase upscaling technique to realistic reservoir cross sections,SPE 51939, presented at the SPE Symposium on Reservoir Simulation,Houston, 1999) found that a constant pressure condition at thesub-domain boundary tends to overestimate flow contributions from highpermeability areas. If the correlation length of permeability is notmuch larger than the grid size, the flow contribution from highpermeability areas is not proportional to the nominal permeabilityratio. The transmissibility between two cells is a harmonic mean that iscloser to the lower permeability. As a result, uniform flux conditionsalong the boundary often yield much better numerical results for asub-domain problem than linear or constant pressure conditions.

Hou and Wu (T. Hou and W. H. Wu, A multiscale finite element method forelliptic problems in composite materials and porous media, J. Comp.Phys, 134:169-189, 1997) also proposed solving a reduced problem$\begin{matrix}{{{\frac{\partial}{\partial x_{t}}\left( {\lambda_{ij}\quad\frac{\partial p}{\partial x_{j}}} \right)_{t}} = 0},} & (11)\end{matrix}$to specify the boundary conditions for the local problem. The subscript_(t) denotes the component parallel to the boundary of the dual coarsecontrol volume 32 or ∂{tilde over (Ω)}. For equation (11) and for thefollowing part of this specification, Einstein summation convention willbe used. The elliptic problem on a control volume {tilde over (Ω)} withboundary conditions of equation (11) on ∂{tilde over (Ω)} can be solvedby any appropriate numerical method. In order to obtain a pressuresolution that depends linearly on the pressures {overscore(p)}^(k)(j=1,4), this preferred embodiment solves four ellipticproblems, one for each cell-center pressure. For instance, to get thesolution for the pressure {overscore (p)}¹ in the coarse grid cellhaving node 1 at its center, {overscore (p)}^(k)=δ_(1k) is set. The foursolutions provide the dual basis functions {tilde over (Φ)}^(k)(k=1,4)in control volume {tilde over (Ω)}, and the pressure solution of thelocal elliptic problem in a control volume {tilde over (Ω)} is thelinear combination $\begin{matrix}{p = {\sum\limits_{k = 1}^{4}{{\overset{\_}{p}}^{k}{{\overset{\sim}{\Phi}}^{k}.}}}} & (12)\end{matrix}$

Accordingly, the flux q across the grid cell interfaces can be writtenas a linear combination $\begin{matrix}{{q = {\sum\limits_{k = 1}^{4}{{\overset{\_}{p}}^{k}q^{k}}}},} & (13)\end{matrix}$where q^(k)(k=1,4) are the flux contributions from the correspondingdual basis functions, given all {tilde over (Φ)}^(k)(k=1,4) from allcontrol volumes {tilde over (106 )}. The effective transmissibilitiesTare computed, which can be used for finite-volume simulations, byassembling the flux contributions, in the preferred embodiment integralflux contributions across the cell interfaces 34.

Note that the domain {tilde over (Ω)} can have any fine-scaledistribution of mobility coefficients λ. Of course the boundarycondition given by equation (11) is an approximation that allows one todecouple the local problems. The MSFV and global fine-scale solutionsare identical, only if equation (11) happens to capture the exactfine-scale pressure solution. However, numerical experiments have beenperformed which indicate that equation (11) is an excellentapproximation of the boundary condition.

Although the MSFV approach is a finite-volume method, it resembles themulti-scale finite-element method of Wu and Hou, briefly mentionedabove. The construction of the dual basis functions is similar, thoughin the present MSFV method they are represented on the dual coarse gridrather than on the boundary of a finite element. A significantdifference is that the present MSFV method is a cell-centeredfinite-volume method and is conservative. On the other hand, the massmatrix in the multi-scale finite-element method is constructed based ona variational principle and does not ensure local conservation. In thenext section, the importance is illustrated of a fine-scale velocityfield that is conservative.

3. Reconstruction of a Conservative Fine-Scale Velocity field

Fluxes across the coarse cell interfaces 34 can be accurately computedby multi-scale transmissibilities T. In some cases, it is interesting toaccurately represent the small-scale velocities u (e.g., to predict thedistribution of solute transported by a fluid). A straightforwardapproach might appear to be to use the dual basis functions {tilde over(Φ)} of equation (12). However, then the reconstructed fine-scalevelocity field is, in general, discontinuous at the cell interfaces ofthe dual grid 30. Therefore, large errors can occur in the divergencefield, and local mass balance is violated. Note that mass conservationis always satisfied for the coarse solution using the present MSFVmethod.

The construction of a second set of local fine scale basis functions Φwill now be described which is fully consistent with the fluxes q acrossthe cell interfaces given by the dual basis functions {tilde over (Φ)}.This second set of fine-scale basis functions Φ allows a conservativefine-scale velocity field to be reconstructed.

FIG. 2 shows a coarse grid 20 with nine adjacent grid cells 22 and acorresponding dual grid 30 of dual coarse control volumes 32 or {tildeover (Φ)}. For indexing purposes, these particular cells andcorresponding dual volumes shall now be identified with numerals “1-9”and letters “A-D” at their respective centers. Also shown is theunderlying fine grid 24 of fine grid cells 26. The coarse grid, havingthe nine adjacent coarse cells 1-9, is shown in bold solid lines. Thecorresponding dual grid 30 of dual coarse control volumes A-D aredepicted with bold dashed lines. The underlying fine grid 24 of finegrid cells 26 is shown with thin dotted lines.

To explain the reconstruction of the fine-scale velocity, the massbalance of the center grid cell 5 is examined. The coarse scale pressuresolution, together with the dual basis functions {tilde over (Φ)},provides the fine-scale fluxes q across the interfaces of coarse cell 5.

To obtain a proper representation of the fine-scale velocity field incoarse cell 5, (i) the fine-scale fluxes across an interface of coarsecell 5 must match, and (ii) the divergence of the fine-scale velocityfield within the coarse volume satisfies $\begin{matrix}{{{\nabla{\cdot u}} = \frac{\int_{\partial{\overset{\_}{\Omega}}_{5}}{q\quad{\mathbb{d}\Gamma}}}{\int_{{\overset{\_}{\Omega}}_{5}}{\mathbb{d}\Omega}}},} & (14)\end{matrix}$where {overscore (Ω)}₅ is the coarse grid cell 5. The fine-scale flux qacross the boundary of grid cell 5 depends on the coarse pressuresolutions in grid cells 1-9. Therefore, the fine-scale velocity fieldwithin coarse grid cell 5 can be expressed as a superposition of finescale basis functions Φ^(i)(i=1,9). With the help of FIG. 3, whichdepicts the needed dual coarse control volumes, the construction of thefine-scale basis functions Φ^(i) will be described. Each coarse cellpressure {overscore (p)}(i=1,9) contributes to the fine-scale flux q.For example, let the contribution of the pressure in cell 2 to the fluxq in grid cell 5 be q⁽²⁾. Note that q⁽²⁾ is composed of contributionsq_(A) ⁽²⁾ and q_(B) ⁽²⁾ coming from the dual basis functions associatedwith node 2 of volume A and volume B, respectively. To compute thefine-scale basis function Φ^(i) associated with the pressure in a coarsecell i, {overscore (p)}^(j)=δ_(ij) is set, and the pressure field isconstructed according to the following equation. $\begin{matrix}{p = {\sum\limits_{k \in {\{{A,B,C,D}\}}}\quad{\sum\limits_{j = 1}^{9}{{\overset{\_}{p}}^{j}\quad{{\overset{\sim}{\Phi}}_{k}^{j}.}}}}} & (15)\end{matrix}$

The fine-scale fluxes q are computed from the pressure field. Thesefluxes provide the proper boundary condition for computing thefine-scale basis function Φ^(i). To solve the elliptic problem∇·(λ·∇p)=ƒ′ on {overscore (Ω)}₅  (16)with the boundary conditions described above, solvability must beensured. This is achieved by setting $\begin{matrix}{{f^{\prime} = \frac{\int_{\partial{\overset{\_}{\Omega}}_{5}}{q\quad{\mathbb{d}\Gamma}}}{\int_{{\overset{\_}{\Omega}}_{5}}{\mathbb{d}\Omega}}},} & (17)\end{matrix}$which is an equally distributed source term within {overscore (Ω)}₅.Finally, the solution of the elliptic problem, (16) and (17), is thefine-scale basis function Φ^(i) for coarse cell 5 associated with thepressure in volume i. The small-scale velocity field is extracted fromthe superposition $\begin{matrix}{p = {\sum\limits_{j = 1}^{9}{{\overset{\_}{p}}^{j}{\Phi_{5}^{j}.}}}} & (18)\end{matrix}$

For incompressible flow, this velocity field is divergence freeeverywhere. Computing the fine-scale basis functions Φ^(i) requiressolving nine small elliptic problems, which are of the same size asthose for the transmissibility calculations. Note that this step is apreprocessing step and has to be done only once. Furthermore, theconstruction of the fine-scale basis functions Φ^(i) is independent andtherefore well suited for parallel computation. The reconstruction ofthe fine-scale velocity field is a simple superposition and is ideallyperformed only in regions of interest.

III. Implemenation of the MSFV Method

FIG. 4 is a flow chart summarizing the steps employed in a preferredembodiment in simulating a reservoir using the MSFV algorithm of thisinvention. The MSFV algorithm consists of six major steps:

A. compute transmissibilities Tfor coarse-scale fluxes (step 100);

B. construct fine-scale basis functions (step 200);

C. compute a coarse solution at a new time level; (step 300);

D. reconstruct the fine-scale velocity field in regions of interest(step 400);

E. solve transport equations (step 500); and

F. recompute transmissibilities and also the fine-scale basis functionsin regions where the total mobility has changed more than apredetermined amount (step 600).

Steps A-D describe a two-scale approach. The methodology can be appliedrecursively with successive levels of coarsening. In cases of extremelyfine resolution, this multi-level approach should yield scalablesolutions. Parts E and F account for transport and mobility changes dueto evolving phases and will be described in more detail below.

A. Computing Transmissibilities for Coarse-Scale Fluxes—Step 100

The transmissibility calculations can be done in a stand alone module(T-module) and are well suited for parallel computation. Thetransmissibilities Tcan be written to a file for use by anyfinite-volume simulator that can handle multi-point flux discretization.

Referring now to FIG. 5, a flowchart describes the individual.stepswhich are undertaken to compute the transmissibilities Tfor a coarsescale model. First, a fine-scale grid having fine cells with anassociated permeability field K are created (step 110). Next, a coarsegrid, having coarse cells corresponding to the fine scale grid, iscreated (step 120). The fine and coarse grids are then passed into atransmissibility or T-module.

Dual coarse control volumes {tilde over (Ω)} are constructed (step 130),one for each node of the coarse grid. For each dual coarse controlvolume {tilde over (Ω)}, dual or coarse scale basis functions Φ_(cs) areconstructed (step 140) by solving local elliptic problems (equation(10)) for each volume {tilde over (Ω)}. This local elliptic problem, asdescribed in section II.A.2 above, and the permeability field Kassociated with the fine grid are used and the boundary conditionscorresponding to equation (11) are utilized (step 135) in solving theelliptic problem. In cases where the fine and coarse grids arenonconforming (e.g., if unstructured grids are used), oversampling maybe applied. Finally, the integral coarse scale fluxes q across theinterfaces of the coarse cells are extracted (step 150) from the dualbasis functions {tilde over (Φ)}. These integral coarse scale fluxes qare then assembled (step 160) to obtain MSFV-transmissibilities Tbetween grid cells of the coarse grid.

The computation of transmissibilities T can be viewed as an upscalingprocedure. That is, the constructed coarse pressure solutions aredesigned to account for, in some manner, the fine-scale description ofthe permeability K in the original fine scale grid model. Thus, partA—step 100—computing transmissibilities, is preferably a separatepreprocessing step used to coarsen the original fine scale model to asize manageable by a conventional reservoir simulator.

These transmissibilities T may be written to a file for later use. Afinite-volume simulator that can handle multi-point flux discretizationcan then use these transmissibilities T.

B. Construction of Fine-Scale Basis Function and Fine Scale VelocityField—Step 200

FIG. 6 is a flowchart describing the steps taken to construct a set offine scale basis functions Φ which can be isolated in a separate finescale basis function Φ module. These fine scale basis functions Φ canthen be used to create a fine scale velocity field. This module is onlynecessary if there is an interest in reconstructing the fine-scalevelocity field from the coarse pressure solution. As described inSection lI.A.3 above, if the original dual basis functions {tilde over(Φ)} are used in reconstructing the fine-scale velocity field, largemass balance errors can occur. Here, steps are described to compute thefine-scale basis functions Φ, which can be used to reconstruct aconservative fine-scale velocity field. The procedure (step 200) of FIG.4 follows the description of Section II.A.3 and has to be performed onlyonce at the beginning of a simulation and is well suited for parallelcomputation.

The fine-scale grid (step 210), with its corresponding permeabilityfield K, the coarse grid (step 220), and the dual basis functions {tildeover (Φ)} (step 230) are passed into a fine scale basis function Φ. Apressure field is constructed from the coarse scale pressure solutionand dual basis functions (step 250). The fine scale fluxes for thecoarse cells are then computed (step 260). For each control volume,elliptic problems are solved, using the fine scale fluxes as boundaryconditions, to determine fine scale basis functions (step 270). Thesmall scale velocity field can then be computed from the superpositionof cell pressures and fine scale basis functions. The results may thenbe output from the module. In many cases, the fine-scale velocity fieldhas to be reconstructed in certain regions only, as will be described infuller detail below. Therefore, in order to save memory and computingtime, one can think of a in situ computation of the fine-scale basisfunctions Φ, which, once computed, can be reused.

C. Computation of the Coarse Solution at the New Time—Step 300

Step 300 can be performed by virtually any multi-point stencilfinite-volume code by using the MSFV-transmissibilities T for the fluxcalculation. These coarse fluxes effectively capture the large-scalebehavior of the solution without resolving the small scales.

D. Reconstruction of the Fine-Scale Velocity Field—Step 400

Step 400 is straight forward. Reconstruction of the fine-scale velocityfield in regions of interest is achieved by superposition of thefine-scale basis functions Φ^(i) as described in section II.A.3, step Babove and as shown in FIG. 6. Of course, many variations of the MSFVmethod can be devised. It may be advantageous; however, thatconstruction of the transmissibilities T and fine-scale basis functionsΦ can be done in modules separate from the simulator.

E. Solving Pressure and Transport Equations

1. Numerical Solution Algorithm—Explicit Solution Multi-phase flowproblems may be solved in two stages. First, the total velocity field isobtained from solving the pressure equation (4), and then the hyperbolictransport equation (5) is solved. To solve the pressure equation, theMSFV method, which has been described above is used. The difference fromsingle phase flow is that in this case the mobility term λ reflects thetotal mobility of both phases, and then the obtained velocity field u isthe total velocity in the domain. The reconstructed fine-scale velocityfield u is then used to solve the transport equation on the fine grid.The values of k_(o,w) are taken from the upwind direction; timeintegration may be obtained using a backward Euler scheme. Note that, ingeneral, the dual and fine scale basis functions ({tilde over (Φ)},Φ)must be recomputed each time step due to changes in the saturation(mobility) field.

2. Numerical Solution Algorithm—Implicit Coupling

In the preferred embodiment of this invention, the MSFV method utilizesan algorithm with implicit calculations. The multi-phase flow problem issolved iteratively in two stages. See FIG. 7 for a diagram of thismethod illustrating the coupling between the pressure and saturationequations.

First, in each Newton step, a saturation field S is established—eitherinitial input or through an iteration (step 510). Next, a pressureequation (see equation (19) below) is solved (step 520) using the MSFVtechniques described above to obtain (step 530) the total velocityfield. Then a transport equation (see equation (20) below) is solved(step 540) on the fine grid by using the reconstructed fine-scalevelocity field u. In this solution, a Schwarz overlap technique isapplied, i.e., the transport problem is solved locally on each coarsevolume with an implicit upwind scheme, where the saturation values fromthe neighboring coarse volumes at the previous iteration level are usedfor the boundary conditions. Once the Schwarz overlap scheme hasconverged (steps 550, 560)—for hyperbolic systems this method is veryefficient—the new saturation distribution determines the new totalmobility field for the pressure problem of the next Newton iteration.Note that, in general, some of the basis functions have to be recomputedeach iteration.

The superscripts n and v denote the old time and iteration levels,respectively. Saturation is represented byS, the total velocity field byu, the computation of the velocity by the operator Π, and thecomputation of the saturation by Σ. The new pressure field p^(v+1) isobtained by solving∇·(k(k _(o)(S ^(v))+k _(w)(S ^(v))∇p ^(v+1))=q _(o) +q _(w),   (19)from which the new velocity field u^(v+1) is computed. The newsaturation field S^(v+1) is obtained by solving $\begin{matrix}{{{\Phi\frac{S^{v + 1} - S^{n}}{\Delta\quad t}} + {\nabla{\cdot \left( {\frac{k_{o}\left( S^{v + 1} \right)}{{k_{o}\left( S^{v + 1} \right)} + {k_{w}\left( S^{v + 1} \right)}^{\prime}}u^{v + 1}} \right)}}} = {- q_{o}}} & (20)\end{matrix}$

F. Recomputing Transmissibilities and Fine-Scale BasisFunctions—Adaptive Scheme

The most expensive part of the MSFV algorithm for multi-phase flow isthe reconstruction of the coarse scale and fine-scale basis functions({tilde over (Φ)},Φ). Therefore, to obtain higher efficiency, it isdesirable to recompute the basis functions only where it is absolutelynecessary. An adaptive scheme can be used to update these basisfunctions. In the preferred exemplary embodiment, if the condition$\begin{matrix}{\frac{1}{{1 +} \in_{\lambda}}\left\langle {\frac{\lambda^{n}}{\lambda^{n - 1}}\left\langle {{1 +} \in_{\lambda}} \right.} \right.} & (23)\end{matrix}$is not fulfilled (the superscripts n and n−1 denote the previous twotime steps and ε_(λ) is a defined value) for all fine cells inside acoarse dual volume, then the dual basis functions of that control volumehave to be reconstructed. Note that condition (23) is true if λ changesby a factor which is larger than 1/(1+ε_(λ)) and smaller than 1+ε_(λ).An illustration of this scheme is shown in FIG. 8, where the fine andthe coarse grid cells are drawn with thin and bold lines, respectively.The black squares represent the fine cells in which condition (23) isnot fulfilled. The squares with bold dashed lines are the controlvolumes for which the dual basis functions have to be reconstructed. Theshaded regions represent the coarse cells for which the fine-scale basisfunctions have to be updated. In the schematic 2D example of FIG. 8,only 20 of 196 total dual basis functions and 117 of 324 totalfine-scale basis functions have to be reconstructed. Of course, thesenumbers depend heavily on the defined threshold ε_(λ). In general, asmaller threshold triggers more fine volumes, and as a consequence morebasis functions are recomputed each time step. For a wide variety oftest cases, it has been found that taking ε_(λ) to be>0.2 yieldsmarginal changes in the obtained results.

IV. Numerical Results

This MSFV method, combined the implicit coupling scheme shown in FIG. 7,has been tested for two phase flow (u_(o)/u_(w)≡10) in a stiff 3D modelwith more than 140,000 fine cells. It has been demonstrated that themulti-scale results are in excellent agreement with the fine-scalesolution. Moreover, the MSFV method has proven to be approximately 27times more efficient than the established oil reservoir simulatorChears. However, in many cases the computational efficiency iscompromised due to the time step size restrictions inherent for IMPESschemes. This problem may be resolved by applying the fully implicitMSFV method, which was described in the previous section. Here numericalstudies show the following:

(1) The results obtained with the implicit MSFV method are in excellentagreement with the fine-scale results.

(2) The results obtained with the implicit MSFV method are not verysensitive to the choice of the coarse grid.

(3) The implicit MSFV for two phase flow overcomes the time step sizerestriction and therefore very large time steps can be applied.

(4) The results obtained with the implicit MSFV method are, to a largeextent, insensitive to the time step size. and

(5) The implicit MSFV method is very efficient.

For the fine-scale comparison runs, the established reservoir simulatorChears was used. The efficiency of both the implicit MSFV method and thefine scale reservoir simulator depends on the choice of variousparameter settings which were not fully optimized.

A. Test Case

To study the accuracy and efficiency of the fully implicit MSFValgorithm, 2D and 3D test cases with uniformly spaced orthogonal 60×220and 60×220×85 grids were used. The 3D grid and permeability field arethe same as for the SPE 10 test case, which is regarded as beingextremely difficult for reservoir simulators. While this 3D test case isused for computational efficiency assessment, the 2D test cases, whichconsist of top and bottom layers, serves to study the accuracy of theMSFV method. FIG. 9 illustrates the 3D test case posed by thepermeability field of the SPE 10 problem. The darker areas indicatelower permeability. An injector well is placed in the center of thefield and four producers in the corners. These well locations are usedfor all of the following studies. The reservoir is initially filled withoil and u_(o)/u_(w)=10 and k_(r) _(o,w) =S_(o) ² _(,w).

B. 2D Simulation of the Top and Bottom Layers

The MSFV simulator used lacked a sophisticated well model. That is,wells are modeled by defining the total rates for each perforated coarsevolume. Therefore, in order to make accuracy comparisons between MSFVand fine-scale (Chears reservoir simulator) results, each fine-scalevolume inside each perforated coarse volume becomes a well in the Chearsruns. For large 3D models, this poses a technical problem since Chearsreservoir simulator is not designed to handle an arbitrary large numberof individual wells. For this reason, it was determined to do anaccuracy assessment in 2D, i.e., with the top and the bottom layers ofthe 3D model. These two layers, for which the permeability fields areshown in FIGS. 10A and 10B, are representative for the twocharacteristically different regions of the full model.

MSFV simulations were performed with uniformly spaced 10×22 and 20×44coarse grids. The results were compared with the fine-scale solution ona 60×220 grid. As in the full 3D test case, there are four producers atthe corners which are distributed over an area of 6×10 fine-scalevolumes. The injector is located in the center of the domain and isdistributed over an area of 12×12 fine-scale volumes. The rates are thesame for all fine-scale volumes (positive for the producer volumes andnegative for the injector volumes). FIGS. 11A-C and 12A-C show thepermeability fields of the respective top and the bottom layers. Theblack is indicative of low permeability. These two layers arerepresentative for the two characteristically different regions of thefull 3D model. FIGS. 11A-C and 12A-C show the computed saturation fieldsafter 0.0933 PVI (pore volume injected) for the top and the bottomlayers, respectively. While FIGS. 11C and 12C show the fine-scalereference solutions, FIGS. 11A and 11B and 12A and 12B show the MSFVresults for 10×22 and 20×44 coarse grids, respectively. For both layers,it can be observed that the agreement is excellent and that themulti-scale method is hardly sensitive to the choice of the coarse grid.A more quantitative comparison is shown in FIGS. 13A and 13B where thefine-scale and multi-scale oil cut and oil recovery curves are plotted.Considering the difficulty of these test problems and the fact that twoindependently implemented simulators are used for the comparisons, thisagreement is quite good. In the following studies, it will bedemonstrated that for a model with 1,122,000 cells, the MSFV method issignificantly more efficient than fine-scale simulations and the resultsremain accurate for very large time steps.

C. 3D Simulations

While 2D studies are appropriate to study the accuracy of the implicitMSFV method, large and stiff 3D computations are. required for ameaningful efficiency assessment. A 3D test case was employed asdescribed above. A coarse 10×22×17 grid, shown in FIG. 14, was used and0.5 pore volumes were injected. Opposed to the MSFV runs, the wells forthe CHEARS simulations were defined on the fine-scale. Table 1 belowshows CPU time and required number of times steps for the CHEARSsimulation and two MSFV runs. TABLE 1 EFFICENCY COMPARISON BETWEEN MSFVand FINE SCALE SIMULATIONS Coarse Recomputed Pressure CPU TIME BasisComputations Simulator (minutes) Time steps Functions (%) (%) Chears3325 790 MSFV 297 200 10 98 MSFV 123 50 26 100

While Chears uses a control algorithm, the time step size in themulti-scale simulations was fixed. It is due to the size and stiffnessof the problem that much smaller time steps have to be applied for asuccessful Chears simulation. The table shows that the implicit MSFVmethod can compute the solution approximately 27 times faster thanCHEARS. FIG. 15 shows the oil cut and recovery curves obtained withmulti-scale simulations using 50 and 200 time steps. The close agreementbetween the results confirms that the method is very robust in respectto time step size. Since the cost for MSFV simulation scales almostlinearly with the problem size and since the dual and fine-scale basisfunction can be computed independently, the method is ideally suited formassive parallel computations and huge problems.

While in the foregoing specification this invention has been describedin relation to certain preferred embodiments thereof, and many detailshave been set forth for purpose of illustration, it will be apparent tothose skilled in the art that the invention is susceptible to alterationand that certain other details described herein can vary considerablywithout departing from the basic principles of the invention.

1. A multi-scale finite-volume method for use in modeling a subsurfacereservoir comprising: creating a fine grid defining a plurality of finecells and having a permeability field associated with the fine cells;creating a coarse grid defining a plurality of coarse cells havinginterfaces between the coarse cells, the coarse cells being aggregatesof the fine cells; creating a dual coarse grid defining a plurality ofdual coarse control volumes, the dual coarse control volumes beingaggregates of the fine cells and having boundaries bounding the dualcoarse control volumes; calculating dual basis functions on the dualcoarse control volumes by solving local elliptic or parabolic problems;extracting fluxes across the interfaces of the coarse cells from thedual basis functions; assembling the fluxes to calculate effectivetransmissibilities between coarse cells; calculating pressure in thecoarse cells using a finite volume method and utilizing the effectivetransmissibilities between coarse cells: and computing a fine-scalevelocity field utilizing the pressure calculated in the coarse cells.2-44. (canceled)